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Chapter 5: Problem 21

Sketch each angle in standard position. (a) \(-30^{\circ}\) (b) \(150^{\circ}\)

Short Answer

Expert verified
The sketch for -30 degrees will be a line from the origin at an angle of 30 degrees to the positive x-axis in the clockwise direction. The sketch for 150 degrees will be a line from the origin at an angle of 150 degrees to the positive x-axis in the counter-clockwise direction.

Step by step solution

01

Sketch for -30 degrees

First, start with the initial side on the positive x-axis, which is the standard position. Since -30 degrees is a negative angle, it will be measured in a clockwise direction from the initial side. Draw a line from the origin forming an angle of 30 degrees with the positive x-axis in the clockwise direction.
02

Label the -30 degrees sketch

After drawing the line, annotate the sketch to indicate that the angle is -30 degrees. At this point, you should have a sketch of an angle at -30 degrees in the standard position.
03

Sketch for 150 degrees

Again, start with the initial side on the positive x-axis, which is the standard position. Since 150 degrees is a positive angle, it will be measured in a counter-clockwise direction from the initial side. Draw a line from the origin forming an angle of 150 degrees with the positive x-axis in the counter-clockwise direction.
04

Label the 150 degrees sketch

After drawing the line, annotate the sketch to indicate that the angle is 150 degrees. At this point, you should have a sketch of an angle at 150 degrees in the standard position.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Position
In geometry, when we talk about angles, it's important to place them in a way that's easy to understand.
This brings us to the concept of the "Standard Position". Standard position is a way to make sure we're all on the same page when sketching or measuring angles.
Here's what you need to know about it:
  • The angle's vertex is at the origin of a coordinate plane.
  • One side of the angle (the initial side) rests along the positive x-axis.
  • The other side (the terminal side) is what's measured to determine the angle's size.
By using standard position, anyone looking at a diagram will know exactly where the angle starts and where it ends.
It's like having a universal starting point, making communication in math much clearer!
Positive Angle
Positive angles are a fundamental concept in understanding angle measurement.
Here's how they work:
  • They are measured starting from the initial side on the positive x-axis.
  • The measurement is made in a counterclockwise direction.
  • Common positive angles include familiar ones like 90掳, 180掳, and 360掳.
This counterclockwise measurement not only helps in defining what a positive angle is, but it also sets the scene for distinguishing between positive and negative angles.
In sketches, a positive angle will look like it's opening upwards from the x-axis toward the top of the page.
Negative Angle
When it comes to negative angles, the clock is essentially turned backwards.
Negative angles are measured differently from positive angles, and here's how:
  • They start from the positive x-axis just like positive angles.
  • The twist is that they're measured in a clockwise direction.
  • Negative angles may not be as intuitive but are just as important in geometry, helping describe motion or orientation in reverse.
For example, a -30掳 angle begins at the positive x-axis and turns downwards.
Negative angles are just another way angles can be utilized to describe a full 360掳 rotation in any direction.
Clockwise and Counterclockwise Measurement
Clockwise and counterclockwise directions are key in understanding how angles are measured.
Let's break it down:
  • Clockwise direction is akin to the movement of a clock's hands, sweeping from the top towards the right, down, and around.
  • In geometry, an angle measured in this direction is negative.
  • On the flip side, counterclockwise direction moves opposite to a clock's hands.
  • An angle measured in this manner is positive.
Understanding these directions helps you visualize and sketch angles accurately.
It's a simple yet important concept, ensuring clarity whether you're drawing an angle or indicating a rotation.

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